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Trigonometric derivatives

The basic trigonometric functions have derivatives that should be memorized: If x is expressed in radians, then:


(sin(x))' = cos(x)  
(cos(x))' = - sin(x)  
(tan(x))' = sec2(x) =  

The Chain Rule

This is a rule for evaluating the derivatives of composite functions


fog = f'(g(x)g'(x)  
  or  
(f (g(x))' = f'(g(x)g'(x)  

For example, the function f (x) = (3x + 2)2 is a composite function where the outer function, f, is a power function (u2), and the inner function, g, is a linear function (3x + 2).

To differentiate this composite function, first treat the inner function as a single variable, and take the derivative of the outer function. Then multiply by the derivative of the inner function:

3x+2 = 23x+2(3)    

Implicit Differentiation

This is a means of finding , the derivative of y with respect to x, even when we do not have a function of the form y = f (x).

Example: Find the slope of the graph at (0, 0) for the following function:

xy2 = x + y    

To solve this problem, we must essentially first find and then plug in the point (0,0) to find the slope.

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